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【正文】 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2 June 3, 2021 73347 Game TheoryLecture 11 17 Exercise of Osborne ? Case 13: check whether there is a mixed strategy in which p11=0, p120, p210, p220, p23=0 (Note this implies p12=1) By theorem 4, we should have 2?p11+2? p12= 3?p11+1? p12 ? 3?p11+2? p12 and p11+p12=1. ? We should have 2?p21+0? p22+1? p23 ?3?p21+1? p22+0? p23 and p21+ p22+ p23 = 1 ? Solve these. If we can get a solution that satisfies p11=0, p120, p210, p220, p23=0 then we have a mixed strategy Nash equilibrium. Otherwise, if we can not find a solution or we find a solution that does not satisfies p11=0, p120, p210, p220, p23=0, then we do not have such a mixed strategy Nash equilibrium. Player 2 L (p21) M (p22) R (p23) Player 1 T (p11) 2 , 2 0 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2 June 3, 2021 73347 Game TheoryLecture 11 18 Exercise of Osborne ? Case 14: check whether there is a mixed strategy in which p11=0, p120, p210, p22=0, p230 (Note this implies p12=1) By theorem 4, we should have 2?p11+2? p12? 3?p11+1? p12 ? 3?p11+2? p12 and p11+p12=1. ? We should have 2?p21+0? p22+1? p23 ? 3?p21+1? p22+0? p23 and p21+ p22+ p23 = 1 ? Solve these. If we can get a solution that satisfies p11=0, p120, p210, p22=0, p230 then we have a mixed strategy Nash equilibrium. Otherwise, if we can not find a solution or we find a solution that does not satisfies p11=0, p120, p210, p22=0, p230, then we do not have such a mixed strategy Nash equilibrium. Player 2 L (p21) M (p22) R (p23) Player 1 T (p11) 2 , 2 0 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2 June 3, 2021 73347 Game TheoryLecture 11 19 Exercise of Osborne ? Case 15: check whether there is a mixed strategy in which p11=0, p120, p21=0, p220, p230 (Note this implies p12=1) By theorem 4, we should have 2?p11+2? p12? 3?p11+1? p12 = 3?p11+2? p12 and p11+p12=1. ? We should have 2?p21+0? p22+1? p23 ? 3?p21+1? p22+0? p23 and p21+ p22+ p23 = 1 ? Solve these. If we can get a solution that satisfies p11=0, p120, p21=0, p220, p230 then we have a mixed strategy Nash equilibrium. Otherwise, if we can not find a solution or we find a solution that does not satisfies p11=0, p120, p21=0, p220, p230, then we do not have such a mixed strategy Nash equilibrium. Player 2 L (p21) M (p22) R (p23) Player 1 T (p11) 2 , 2 0 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2 June 3, 2021 73347 Game TheoryLecture 11 20 Summary ? How to solve Exercise of Osborne ? Next time ? Dynamic games of plete information ? Reading lists ? Cha of Gibbon
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